Asymptotic theory for static behavior of elastic anisotropic I-beams

End effects for prismatic anisotropic beams with thin-walled, open cross-sections are analyzed by the variational-asymptotic method. The decay rates for disturbances at the ends of prismatic beams are evaluated, and the most influential end disturbances are incorporated into a refined beam theory. Thus, the foundations of Vlasovs theory, as well as restrictions on its applicability, are obtained from the variational-asymptotic point of view. Vlasovs theory is proved to be asymptotically correct for isotropic I-beams. The asymptotically correct generalization of Vlasovs theory for static behavior of anisotropic beams is presented. In light of this development, various published generalizations of Vlasovs theory for thin-walled anisotropic beams are discussed. Comparisons with a numerical 3-D analysis are provided, showing that the present approach gives the closest agreement of all published theories. The procedure can be applied to any thin-walled beam with open cross-sections.     

Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams

Large amplitude periodic forced vibration of curved beams under periodic excitation is investigated using a three-noded beam element. The element is based on the higher-order shear deformation theory satisfying interlayer continuity of displacements and transverse shear stress, and top-bottom conditions on the latter. The periodic responses are obtained using shooting technique coupled with Newmark time marching and arc length continuation algorithm developed. The second order governing differential equations of motion are solved without transforming to the first order differential equations thereby resulting in a computationally more efficient algorithm. The effects of excitation amplitude, support conditions and beam curvature on the frequency versus response amplitude relation are highlighted. The typical frequency response curves for isotropic and cross-ply laminated curved beams are presented. Phenomenon of strong modal interactions is observed.     

A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

A microscale nonlinear Bernoulli–Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton’s principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Bending–stretching analysis of moderately thick functionally graded anisotropic wide beams

This paper presents an analytical solution for static analysis of moderately thick laminated composite wide beams whose fiber orientation angle varies continuously through the thickness direction. Since these anisotropic beams have such a monoclinic stiffness matrix form, the strain components which are ignored for isotropic wide beams must be taken into account. To this end, a refined displacement field taking into account the entire shear strains is used. The equilibrium equations are obtained and solved analytically for beams with different boundary conditions. Stress and displacement components of the functionally graded (FG) beam are obtained, and the effects of FG parameter, boundary condition and length–thickness ratio are studied.     

Nonlinear vibration of an axially loaded beam carrying multiple mass–spring–damper systems

This paper deals with the nonlinear vibration of a beam subjected to a tensile load and carrying multiple spring–mass–dashpot systems. The nonlinearity is attributable to mid-plane stretching, damping, and spring constant. Explicit expressions are presented for the frequency equations, mode shapes, nonlinear frequency, and modulation equations. The validity of the results is demonstrated via comparison with results in the literature. Parametric studies are conducted on beams with varying boundary conditions to investigate the effect of the location and magnitude of the spring–mass–dashpot system, as well as the role of the tension.     

Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings

An efficient scheme, called quasi-linearization finite differences, is developed for large-deflection analysis of prismatic and non-prismatic slender cantilever beams subjected to various types of continuous and discontinuous external variable distributed and concentrated loads in horizontal and vertical global directions. Simultaneous equations of highly nonlinear and linear terms are obtained when casting the derived exact highly nonlinear governing differential equation using central finite differences on the nodes along the beam. A quasi-linearization scheme is used to solve these equations based on successive corrections of the nonlinear terms in the simultaneous equations. The nonlinear terms in the simultaneous equations are assumed constant during each correction (iteration). Several representative numerical examples of prismatic and non-prismatic slender cantilever beams with different loading conditions are analyzed to illustrate the merits of the adopted numerical scheme as well as its validity, accuracy and efficiency. The results of the present scheme are checked using large-displacement finite element analysis by the MSC/NASTRAN program. A comparison between the present secheme, MSC/NASTRAN and available results from the literature reveals excellent agreement. The advantage of the new scheme is that the load can be applied in one step with few iterations (3–6 iterations).     

Nonlinear damping in large-amplitude vibrations: modelling and experiments

Experimental data clearly show a strong and nonlinear dependence of damping from the maximum vibration amplitude reached in a cycle for macro- and microstructural elements. This dependence takes a completely different level with respect to the frequency shift of resonances due to nonlinearity, which is commonly of 10–25% at most for shells, plates and beams. The experiments show that a damping value over six times larger than the linear one must be expected for vibration of thin plates when the vibration amplitude is about twice the thickness. This is a huge change! The present study derives accurately, for the first time, the nonlinear damping from a fractional viscoelastic standard solid model by introducing geometric nonlinearity in it. The damping model obtained is nonlinear, and its frequency dependence can be tuned by the fractional derivative to match the material behaviour. The solution is obtained for a nonlinear single-degree-of-freedom system by harmonic balance. Numerical results are compared to experimental forced vibration responses measured for large-amplitude vibrations of a rectangular plate (hardening system), a circular cylindrical panel (softening system) and a clamped rod made of zirconium alloy (weak hardening system). Sets of experiments have been obtained at different harmonic excitation forces. Experimental results present a very large damping increase with the peak vibration amplitude, and the model is capable of reproducing them with very good accuracy.     

Modal analysis and dynamic responses of a rotating Cartesian manipulator with generic payload and asymmetric load

Abstract

Here, investigation to explore the effect of generic payload and externally applied asymmetric load on the calculation of modal parameters and dynamic performance of a rotating flexible manipulator under prismatic motion has been established. We thus have developed a dynamic model of a rotating Cartesian manipulator with a payload whose center of gravity doesn’t coincide with the point of attachment, to determine the modal parameters i.e., natural frequency and corresponding mode-shape. These modal parameters are then illustrated graphically upon varying parameters like offset parameters (i.e., offset mass, offset inertia, offset length), mass and stiffness of rotary actuator, and amplitude and frequency of asymmetric load. An investigation into the nonlinear dynamics of the system accounting of geometric nonlinearity has been executed while obtained results have been validated numerically within the permissible error at the assorted critical points in frequency characteristic curves. Current research further investigates the influences of offset parameters, mass and stiffness of the actuator, frequency and amplitude of axial force on the steady state responses for the primary and sub-harmonic resonance conditions to reveal the built-in saddle-node and pitchfork bifurcation due to which the system losses its structural stability. This work enables an insight into the modal characteristics and nonlinear behavior of a rotating-Cartesian manipulator with a generic payload under asymmetric axial force and prismatic motion.     

各向异性非线性强度条件下的边坡稳定性

解释了土体强度各向异性、非线性的物理本质,结合常规直剪试验、三轴试验结果,在前人工作基础上建立了边坡稳定性分析中强度各向异性、非线性的描述方法,其中特别提出了一个各向异性函数．基于Janbu普遍条分法(GPS),运用SPREADSHEET模板程序,提出了一个能将各向异性、非线性强度准则逐点等效到Mohr-Coulomb直线强度准则处理上的迭代方法,准确方便地获得了非线性强度下的边坡稳定性分析．最后的算例展示了方法的使用过程。     

Elasticity solutions for plane anisotropic functionally graded beams

This paper considers the plane stress problem of generally anisotropic beams with elastic compliance parameters being arbitrary functions of the thickness coordinate. Firstly, the partial differential equation, which is satisfied by the Airy stress function for the plane problem of anisotropic functionally graded materials and involves the effect of body force, is derived. Secondly, a unified method is developed to obtain the stress function. The analytical expressions of axial force, bending moment, shear force and displacements are then deduced through integration. Thirdly, the stress function is employed to solve problems of anisotropic functionally graded plane beams, with the integral constants completely determined from boundary conditions. A series of elasticity solutions are thus obtained, including the solution for beams under tension and pure bending, the solution for cantilever beams subjected to shear force applied at the free end, the solution for cantilever beams or simply supported beams subjected to uniform load, the solution for fixed–fixed beams subjected to uniform load, and the one for beams subjected to body force, etc. These solutions can be easily degenerated into the elasticity solutions for homogeneous beams. Some of them are absolutely new to literature, and some coincide with the available solutions. It is also found that there are certain errors in several available solutions. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a functionally graded anisotropic cantilever beam.     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

强电场和机械荷载联合作用下压电层合梁的非线性变形

本文研究简支,固支和悬臂压电层合梁在强电场和机械荷载联合作用下的非线性变形。考虑材料的电致伸缩和电致弹性压电效应以及几何非线性导出压电层合梁的数学模型。并求得在电场和均布力联合作用下各种边界条件梁的挠度和位移解析表达式。通过对双压电晶片梁和单压电晶片梁的数值计算及分析得到线性与非线性模型之间的差别和适用范围。     

Geometrical nonlinear analysis of thin-walled composite beams using finite element method based on first order shear deformation theory

Based on a seven-degree-of-freedom shear deformable beam model, a geometrical nonlinear analysis of thin-walled composite beams with arbitrary lay-ups under various types of loads is presented. This model accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The general nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed to solve the problem. Numerical results are obtained for thin-walled composite beam under vertical load to investigate the effects of fiber orientation, geometric nonlinearity, and shear deformation on the axial–flexural–torsional response.     

Nonlinear lateral vibrations and two-to-one resonant responses of a single pile with soil-structure interaction

Nonlinear responses of a single pile with soil-structure interaction under primary resonance and two-to-one internal resonance are investigated. Considering the effect of soil-structure interaction and geometric nonlinearity, the Hamilton principle is applied to derive the equation of motion of the pile subjected to a lateral excitation and axial load. Then, the effective nonlinear coefficient and the frequency-response equation are presented by employing the multimode discretization and the method of multiple scales. Moreover, the nonlinear frequencies are obtained for the analysis of the nonlinear dynamic characteristics of the pile. The equilibrium and dynamic solutions of modulation equations are examined by using the Newton-Raphson, shooting, and continuation methods. The nonlinear responses of the pile are explored by means of the backbone curves and the frequency (force)-response curves. Meanwhile, the effect of the axial load on the nonlinear dynamic characteristics of the pile is discussed. The numerical results show that the nonlinear responses of the pile exhibit some rich and complex nonlinear phenomena.     

THEORY OF ELASTICITY OF AN ANISOTROPIC BODY FOR THE BENDING OF BEAMS

A theory of elasticity for the bending of orthogonal anisotropic beams has been developed by analogy with the special case, which can be obtained by applying the theory of elasticity for bending of transversely isotropic plates to the problems of two deminsions. In this paper, we present a method to solve the problems of bending of orthogonal anisotropic beams and a new theory of the deep-beam whose ratio of depth to length is larger. It is pointed out that Reissner's theory to account for the effect of transverse shear deformation is not very approximate in the components of stress,     

适用于几何非线性气动弹性分析的结构动力学降阶模型方法研究

几何非线性是壁板颤振和大展弦比机翼气动弹性等问题的一个主要特征,在进行数值仿真分析时往往需要采用商业非线性有限元求解器,存在计算量大和耦合迭代策略不易控制等问题。本文发展了一种适用于几何非线性的结构动力学降阶模型(CSD-ROM),利用广义坐标的非线性多项式表征非线性内力,采用参数识别方法获取多项式系数,并通过增加额外的线性模态来改善模型预测精度。基于此方法,分别针对壁板颤振、切尖三角翼的CFD/CSD-ROM非线性颤振问题开展了时域响应分析。计算结果表明,通过CSD-ROM计算出的壁板颤振速度为590 m/s,颤振频率为174 Hz,与有限元结果误差分别为0.8%和1.7%。马赫数0.879时切尖三角翼的颤振动压预测结果为2.25 psi,与非线性有限元相比的误差为3.8%。本文采用的非线性和线性模态基底组合方法,在保证计算精度的基础上可有效降低训练样本数量,一定程度上可替代非线性有限元开展气动弹性分析。     

Affine transformations of the equations of the linear theory of elasticity

This paper deals with the general formulas of affine transformations that preserve invariance of the static equations of the linear theory of elasticity in the case of arbitrary anisotropic materials. The invariance of the equations with respect to affine transformations allows one to model a given anisotropic material by another material. All anisotropic materials are divided into classes of mutually congruent materials. The congruency conditions are obtained for orthotropic and isotropic materials and for orthotropic and transversely isotropic materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 124–134, July–August, 2006.     

采用瓣叶复合材料的生物心脏瓣膜的力学分析

采用微观组织结构分析及宏观复合材料分析结合的方法,分析了猪主动脉瓣的非线性复合材料性质,提出了一种适用于猪主动脉瓣的非线性复合材料本构模型,用提出的非线性复合材料本构模型,对闭合承载状态下的等厚度与变厚度几何模型的猪主动脉瓣的应力分布及变形进行了有限元数值模拟,发现：与各向同性瓣叶相比,单向增强复合材料的瓣叶不但具有较强的承载能力,而且具有较大的柔软性。     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性，同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应，利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析，在不同的条件下，该方程在相平面上存在同宿轨道或异宿轨道， 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法，对该非线性方程进行 求解，得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解，讨论了这些解存 在的必要条件，这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程，从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.